Evaluate the integrals in Exercises 53–58.
∫ from 0 to π/2 of sin(x) cos(x) dx

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Recognize that the integral is \( \int_0^{\frac{\pi}{2}} \sin(x) \cos(x) \, dx \). This is a definite integral with limits from 0 to \( \frac{\pi}{2} \).

Use a trigonometric identity or substitution to simplify the integrand. One useful identity is \( \sin(2x) = 2 \sin(x) \cos(x) \), so \( \sin(x) \cos(x) = \frac{1}{2} \sin(2x) \).

Rewrite the integral using this identity: \( \int_0^{\frac{\pi}{2}} \sin(x) \cos(x) \, dx = \int_0^{\frac{\pi}{2}} \frac{1}{2} \sin(2x) \, dx \).

Factor out the constant \( \frac{1}{2} \) from the integral: \( \frac{1}{2} \int_0^{\frac{\pi}{2}} \sin(2x) \, dx \).

Perform a substitution to integrate \( \sin(2x) \): let \( u = 2x \), then \( du = 2 dx \) or \( dx = \frac{du}{2} \). Change the limits accordingly and integrate \( \sin(u) \) with respect to \( u \).

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Definite Integrals

A definite integral calculates the net area under a curve between two specific limits. It is represented as ∫ from a to b of f(x) dx, where a and b are the lower and upper bounds. Evaluating definite integrals involves finding the antiderivative and then applying the Fundamental Theorem of Calculus.

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Trigonometric identities are equations involving trigonometric functions that hold true for all values in their domains. For example, the product-to-sum identity sin(x)cos(x) = (1/2)sin(2x) simplifies integration by converting products into sums, making the integral easier to evaluate.

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This theorem links differentiation and integration, stating that if F is an antiderivative of f on [a, b], then ∫ from a to b of f(x) dx = F(b) - F(a). It allows us to evaluate definite integrals by finding antiderivatives and computing their difference at the bounds.

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